A bomb is projected from the ground at a horizontal range of R. If the bomb explodes mid-air, then the range of its centre of mass is:

1.  $\frac{\mathrm{R}}{2}$

2.  $\mathrm{R}$

3.  $2\mathrm{R}$

4.  $\frac{2\mathrm{R}}{3}$

The law of conservation of angular momentum is valid when:

A man hangs from a rope attached to a hot-air balloon. The man's mass is greater than the mass of the balloon and its contents. The system is stationary in the air. If the man now climbs up to the balloon using the rope, the centre of mass of the "man plus balloon" system will:

Two loads ${\mathrm{P}}_1$ $\mathrm{and}$ ${\mathrm{P}}_2$ (${\mathrm{P}}_1$ $>{\mathrm{P}}_2$) are connected by a string passing over a fixed pulley. The center of gravity of loads are initially at the same height. Find the acceleration of the center of gravity of the system:
1. \(\left(\frac{(P_1-P_2)^{\frac{1}{2}}}{P_1+P_2}\right)\text{g}\)
2. \(\left(\frac{P_1-P_2}{P_1+P_2}\right)\text{g}\)
3. \(\left( \frac{P_1-P_2}{P_1+P_2}\right)^2 \text{g}\)
4. \(\left( \frac{P_1+P_2}{P_1-P_2}\right)\text{g}\)

A rod is falling down with constant velocity \(V_0\) as shown. It makes contact with hinge A and rotates around it. The angular velocity of the rod just after the moment when it comes in contact with hinge A is:

A particle rotating on a circular path of the radius $\frac{4}{\mathrm{\pi }}\mathrm{m}$ at 300 rpm reaches 600 rpm in 6 revolutions. If the angular velocity increases at a constant rate, find the tangential acceleration of the particle.

1. 10 m/s²

2. 12.5 m/s²

3. 25 m/s²

4. 50 m/s²

Four-point masses each of value \(m\) are placed at the corners of a square ABCD of side \(l\). The moment of inertia of this system about an axis passing through A and parallel to BD will be:

A uniform cubical block of side L rests on a rough horizontal surface with coefficient of friction $\mu$. A horizontal force F is applied on the block as shown. If there is sufficient friction between the block and the ground, then the torque due to normal reaction about its centre of mass is:

1.  $Zero$

2.  $FL$

3.  $\frac{FL}{2}$

4.  $\frac{3FL}{2}$

The mass per unit length of a non-uniform rod of length L is given by $\mathrm{\mu }={\mathrm{\lambda x}}^2$ where $\mathrm{\lambda }$ is a constant and x is the distance from one end of the rod. The distance between the centre of mass of the rod and this end is:

1. $\frac{\mathrm{L}}{2}$

2. $\frac{\mathrm{L}}{4}$

3. $\frac{3\mathrm{L}}{4}$

4. $\frac{\mathrm{L}}{3}$

Particles A and B are separated by 10 m, as shown in the figure. If A is at rest and B started moving with a speed of 20 m/s then the angular velocity of B with respect to A at that instant is: